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      Questions tagged [stochastic-processes]

      A stochastic process is a collection of random variables usually indexed by a totally ordered set.

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      Uniform law of large number in a Markov decision process setting?

      Consider $$ R =\sup_{f\in\mathcal{F}} \left[ \frac{1}{n}\sum_{i=1}^n f(X_i) - \mathbb{E}[f(X)] \right] $$ If $X_i$'s are i.i.d., then uniform law of large number shows that, if $\...
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      Characterization of Time-homogeneous flows for conditional expectation

      Let $X_t,Y_t$ be $\mathbb{R}^d$-valued processes. It is well known that for every $t\geq 0$, and every bounded function $\phi:\mathbb{R}^d\rightarrow \mathbb{R}$, there exists a Borel function $f_t:\...
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      71 views

      Feynman-Kac formula for domains with boundary

      As far as I know (I am not an expert), any solution of the heat equation on $\mathbb{R}^n$ or on a closed Riemannian manifold with the given initial condition can be presented in terms of stochastic ...
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      36 views

      Stochastic Fixpoint Approximations of Contractions

      Context/Introduction Consider a contraction $f\colon\mathbb{R}^S\to\mathbb{R}^S$ with $f(X^*)=X^*$ where function evaluation at a certain position is only possible with some stochastic error. Where Y(...
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      61 views

      Skorokhod representation for weak convergence of exchangeable arrays

      Let $(X_e)=(X_e)_{e\in \mathbb{N}^{(k)}}$ be a $k$-dimensional exchangeable real random array (see this note for the definition), where $k\in \{1,2,\ldots \}$ is fixed and $\mathbb{N}^{(k)}$ denotes ...
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      35 views

      A 2d recurrence equation representing a step-free continuous-time Markov process on N^2 with frequency-dependent rates

      $\textbf{Background:}$ Consider a particle in $\mathbb{N}^2$ starting at a point $(x,y)$ that can only move one step at a time along the lattice. The particle moves each up or down in continuous time ...
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      Reference: Stochastic Filtering Infinite Dimensions

      I've come across these Hilbert Space Signal Finite Dimensional Measurements and Linear Gaussian Hilbert space signal and measurements. Is there any literature solving the Zakai equation when both ...
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      Why control a continuous approximation of stochastic gradient descent instead of just the SGD?

      In "Stochastic modified equations and adaptive stochastic gradient algorithms" (Li et. al 2015) the authors approximate stochastic gradient descent, as in $$x_{k+1} = x_k - \eta u_k \nabla f_{\...
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      32 views

      Asymptotic upper densities in infinite binary stochastic processes

      Consider an infinite binary process $X=X_1,X_2,\ldots$ (with corresponding probability $P$). For some bits $1$ is less probable than $0$. I am interested in the following asymptotic upper density : $$\...
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      60 views

      Domain of the Generator of a Bessel process

      Consider the Bessel Process of index $\nu\in (-1,0)$, or dimension $\delta=2\nu-1$ \begin{align} \rho_{t}=x+\frac{\delta-1}{2}\int_{0}^{t}\frac{1}{\rho_{s}}\,ds+W_{t} \end{align} where $(W_{t})_{t\geq ...
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      41 views

      Mean first-passage time for a marked Poisson process

      Given a marked Poisson process in one dimension $$ Y(t)=\sum_{\{t_i,a_i\}}g(t-t_i,a_i) $$ so that $Y(t)$ is a sum of impulses arriving as a Poisson process and the impulses $g$ belong to a family of ...
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      42 views

      Solving a non-linear differential equation, generalized Feynman-Kac formula

      I want to solve the following differential equation $f_t(t,\Gamma)+ A(t)f(t,\Gamma) + B(t)f_\Gamma(t,\Gamma)+ \frac{1}{2}tr(\kappa^\intercal \kappa f_{\Gamma\Gamma}(t,\Gamma))+C(t)f_\Gamma (t,\Gamma)\...
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      35 views

      Stochastic process for minimize a mean

      I have the next problem: consider an inventory process $\{X_{k}\}$ such that $X_{k+1}=X_k+A_k-\xi$, $X_0=25$, where $A_k$ is the number of items of items produced at the $k$th-month and $\xi$ is the ...
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      97 views

      “Brownian motion” related to the $p$-Laplace operator

      The link between the Brownian motion and the Laplace operator is well-known. What stochastic process plays an analogous role with respect to the $p$-Laplace operator?
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      247 views

      How are the fields of dynamical systems, stochastic processes and additive combinatorics, inter-related?

      Currently I’m interested in a couple of fields, namely dynamical systems, stochastic processes, and additive combinatorics. I was wondering if it’s feasible to keep pursuing all 3, and whether I can ...

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